More than 60 years ago, atmospheric scientist Charles David Keeling began regular measurements of carbon dioxide concentrations in the atmosphere. In the heart of the Pacific and far from the largest human sources of the gas, Hawaii's Mauna Loa Observatory was an ideal location for these measurements. Within just two years, Keeling had detected two patterns in the data. The first was an annual rise and fall as the seasons came and went. But the second — a year-by-year increase — suggested something alarming: a rise in carbon dioxide produced by the widespread burning of fossil fuels. In 1965, Keeling's measurements were incorporated into a report for U.S. President Lyndon B. Johnson that described carbon dioxide from fossil fuels as "the invisible pollutant" and warned of its dangers.

Since then, global emissions of carbon dioxide and other greenhouse gases have continued to rise, as have the concerns over the changes that such an atmospheric shift brings.

Observations are still taken at Mauna Loa today, and the resulting "Keeling Curve" reveals that atmospheric carbon dioxide levels have increased by almost a third since the first measurements were taken. The world's average temperature has already warmed by around 1 degree Celsius (1.8 degrees Fahrenheit) since preindustrial times, driving increases in everything from sea levels to the frequency of extreme weather events.

Continuous measurements of carbon dioxide levels at the Mauna Loa Observatory have revealed that humans are contributing to a relentless increase. But these measurements only reveal the portion of emissions that remain in the atmosphere, and not the total emitted by human activities. | (SCRIPPS INSTITUTION OF OCEANOGRAPHY SIO)

For those groups and nations striving to limit global warming, accurately tracking carbon emissions will be key to assessing progress and validating international agreements. But how do scientists do that? And how does the amount released into the air relate to what scientists end up measuring at outposts such as Mauna Loa?

Here's the current state of counting carbon, explained.

Why is monitoring global carbon emissions important?

A comprehensive tally of carbon released is essential not just for assessing which countries are pulling their weight and meeting agreed targets. It's also key to improving understanding of carbon's natural cycle and to more precisely quantifying the link between humankind's emissions and the planet's temperature. But calculating, much less measuring, global carbon dioxide emissions remains an immense technical challenge, since almost every human activity is implicated in the molecule's release.

In Paris in 2015, most of the world reached an agreement on climate change. The deal was to limit the world's warming to well below 2 degrees Celsius (3.6 degrees Fahrenheit), with a target of just 1.5 degrees Celsius (2.7 degrees Fahrenheit) above preindustrial levels. Nations pledged to cut their emissions, and the Paris Agreement aims to periodically review progress. While these pledges are insufficient to achieve the deal's targets, the hope is that countries will gradually ramp up their ambitions, and further ramp down emissions.

The emissions pledges are exactly that — pledges. They are not legally binding, and if a country misses its intended targets, the only diplomatic consequence would be the judgment of the international community.

But all of this relies on a clear picture of the country's emissions in the first place. It's a crucial undertaking, because "monitoring emissions is directly at the heart of the pledge-and-review concept," says Gabriel Chan of the Humphrey School of Public Affairs at the University of Minnesota, who reviewed the state of international climate policy in the Annual Review of Resource Economics.

Atmospheric observations — like those carried out at Mauna Loa — provide a global, cumulative picture, but cannot be decomposed into year-by-year national contributions. Global measures also fail to account for the natural carbon "sinks" — the portion of carbon dioxide emissions that are taken up by the oceans and land. For a clear picture of national emissions, researchers have to start from the bottom up.

How do you calculate a country's carbon emissions?

In theory this is just a matter of math, but in practice it's a question of huge-scale accounting. To get a picture of the carbon dioxide a country emits by burning fossil fuels, all energy use must first be counted. These assessments are already carried out for economic reasons and include tabulating the quantities of different fuels — such as coal, gas, or kerosene — that are produced, traded, converted, or used by a country across all sectors. While the contribution of large sources such as power plants can be relatively straightforward to assess, other ledger entries — such as household activities — "are very hard to account for," says Chan. Accurately estimating these sources requires surveys to assess what goes on within a typical home and extrapolating from those.

Figures from these energy assessments can be used to estimate national carbon dioxide emissions. Inventories provided by the Intergovernmental Panel on Climate Change list the amount of carbon dioxide released when an amount of a particular fuel is burned. These "emissions factors" can be combined with energy data to calculate the amount of carbon dioxide that will be released from all of a nation's fossil fuel combustion.

The International Energy Agency, which has been collecting energy data for over 40 years and calculates its own statistics on emissions, recognizes the difficulty in getting it right. "We do really spend a substantial part of our time validating the data," says Roberta Quadrelli of the IEA's Energy Data Centre. For example, if a refinery disappears from the data, it's essential to find out whether its absence was caused by the refinery closing or by being missed in the reporting.

Issues can also crop up when converting energy use to emissions. A 2015 study found that in one year China's emissions had been overestimated by some 14 percent. "The error bar was like an entire Germany," says Chan. This huge miscalculation was primarily caused by a misassessment of the quality of the coal burned in Chinese power plants. Given the scale of China's emissions (currently higher than those of any other nation, although not on a per person basis), some errors are not a surprise, says climate scientist Corinne Le Quéré, who leads the annual Global Carbon Budget report. "I don't want to give them excuses, but it's a big challenge."

What about tracking smaller, less obvious sources of carbon?

In fact, one of the biggest challenges in tracking carbon dioxide emissions isn't related to burning fossil fuels at all. Certain changes in land use — such as deforestation or urbanization — can lead to an uptick in carbon dioxide entering the atmosphere through a number of complex processes. An area of much current research, these factors are far harder to assess than emissions from transportation or power plants. And while land use changes were estimated to be responsible for only around 12 percent of global emissions in 2016, they remain a major source of uncertainty about how much carbon is entering the atmosphere.

For all these reasons and more, overall uncertainty in total carbon emissions remains high, equivalent to nearly 10 percent of the calculated annual emissions and more than the European Union's entire fossil fuel emission tally for 2017.

How current are carbon emissions numbers?

Timing poses another challenge. The complexity of tabulating national emission totals also causes delays in reporting. These delays can make a big difference for policy. Official statistics may take many months to appear, meaning negotiators are often working with outdated information, says Niklas Höhne, who founded Climate Action Tracker, which monitors nations' climate commitments and actions.

At the extreme end, during the Copenhagen climate negotiations in 2009, negotiators were working with an IPCC report published in 2007. The report included emissions only up to 2004, and this chain of delays meant that there was a half-decade gap between policy and reality. These five years — it was later shown — had seen a significant increase in emissions, and the scenarios in terms of emissions and temperature targets sketched out in the negotiations were misaligned with the real world. Even as they were unveiled, they were out of date.